Symmetric 18s

Everything about Sudoku that doesn't fit in one of the other sections

Re: Symmetric 18s

Postby eleven » Sat Oct 22, 2011 11:42 am

Thanks olimpia, for the adulation, i am happy, that you like it.

When i'll have time again, i want to make a similar thing with digit symmetrical puzzles. This might be more in Ron's sense.
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Re: Symmetric 18s

Postby olimpia » Sat Oct 22, 2011 1:15 pm

ronk wrote:Depends upon whether one sees them as a set of symmetric patterns with valid puzzles ... or as a set of puzzles, one for each symmetric pattern.

Oh, good point. And that makes sense too, because then the graphic would be a complete collection of shapes (as far as we know), rather than a sampling of sudokus.

I doubt that any of the puzzles are unique to their pattern, so I lean towards the former.

I suspect some of these shapes might have only a single sudoku. Eleven once calculated there are 362 total puzzles (at 105 known shapes) so that would be about 3.4 sudokus per shape. But some shapes had dozens, so maybe a few just have one?

BTW, I had second thoughts on the idea of sending eleven's art into outer space. This would be like telling them the answer to a tricky problem. It would be better to just send a blank grid and see if they send anything back 8-)
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Re: Symmetric 18s

Postby eleven » Sat Oct 22, 2011 5:28 pm

Since i had transformed and reordered the puzzles in the second picture, it is not trivial to find the original ones. E.g. a nice shape was "#57. Ano1", which mutated to the 8th puzzle in the second row then.
Here is a list of the puzzles in VS18colorB.jpg:
Hidden Text: Show
Code: Select all
Row 1:                        Row 5:                       Row 9:                         
                              #44. mauricio                #32. mauricio                 
#62. Ano1                     #105.eleven                  #39. mauricio                 
#6.  mauricio                 #3.  mauricio                #40. mauricio                 
#13. VicReinhart,Odyssey      #24. VicReinhart,Lulin       #96. olimpia,pattern96         
#18. tarek                    #59. Ano1                    #1.  mauricio                 
#61. Ano1                     #55. Ano1                    #101.eleven           
#68. Ano1                     #60. Ano1                    #81. Ano1             
#77. Ano1                     #14. VicReinhart,Electra     #46. mauricio,spaceship       
#103.eleven                   #48. RicoAlan,Jade           #19. olimpia           
Row 2:                        Row 6:                       Row 10:                       
#42. mauricio                 #21. RicoAlan,R828-S12       #10. olimpia           
#12. VicReinhart,Raphael      #22. mauricio                #25. olimpia           
#71. Ano1                     #20. mauricio                #49. olimpia           
#89. olimpia,pattern89        #23. RicoAlan,R828-S10       #70. Ano1             
#9.  RicoAlan,R828-S11        #51. olimpia                 #80. Ano1             
#15. mauricio                 #64. Ano1                    #74. Ano1             
#99. eleven                   #66. Ano1                    #82. Ano1             
#57. Ano1                     #98. olimpia                 #27. olimpia           
#78. Ano1                     #45. VicReinhart,Oaxaca      #83. Ano1             
Row3:                         Row 7:                       Row 11:                       
#84. Ano1                     #16. RicoAlan,R828-S09       #67. Ano1             
#17. VicReinhart,ElMorro      #54. olimpia                 #50. olimpia           
#31. mauricio                 #79. Ano1                    #85. Ano1             
#69. Ano1                     #87. olimpia,pattern87       #56. Ano1             
#91. olimpia,pattern91        #53. olimpia                 #72. Ano1             
#92. olimpia,pattern92        #47. RicoAlan,R747-S13       #88. olimpia,pattern88         
#75. Ano1                     #33. mauricio                #104.eleven           
#94. olimpia,pattern94        #58. Ano1                    #73. Ano1             
#65. Ano1                     #4.  mauricio                #30. mauricio                 
Row 4:                        Row 8:                       Row 12:                       
#95. olimpia,pattern95        #36. mauricio                #34. mauricio                 
#102.olimpia                  #63. Ano1                    #35. mauricio                 
#26. olimpia                  #76. Ano1                    #41. mauricio                 
#43. olimpia                  #86. olimpia,pattern86       #90. olimpia,pattern90         
#7.  mauricio                 #100.eleven                  #52. olimpia           
#5.  mauricio                 #37. mauricio                #29. mauricio                 
#8.  olimpia                  #106.olimpia                 #93. olimpia,pattern93         
#11. mauricio,heart           #28. mauricio                #97. olimpia,pattern97   
#38. mauricio                 #2.  mauricio             

Note, that you can perform 6^5 equivalence operations on such a pattern, so a shape is all but fixed (though you might get many duplicates).
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Re: Symmetric 18s

Postby eleven » Mon Oct 24, 2011 10:48 am

To give an impression of the variety of equivalent shapes i randomly selected 108 out of the 2244 (symmetry-keeping) equivalents of "#1 mauricio". So it would take some time, if you want to select a favorite shape for all representatives.
http://imageshack.us/photo/my-images/545/mauricio1.jpg/
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Re: Symmetric 18s

Postby olimpia » Mon Oct 24, 2011 11:42 pm

Talking about re-arranging shapes, I've wondered if any of the 17 clue puzzles can be re-arranged to be nearly symmetric (on a vertical axis), so that it is only one clue away from complete symmetry. Then an 18th clue can be added, making an 18 clue non-minimal symmetric sudoku. I suspect this doesn't exist, but if anyone has the programming skills to check, it might be interesting to find out. Also, I'm not sure if checking for "near-symmetry" would significantly increase the task compared to a simpler "symmetry" check.

I don't even have a list of the nearly 50,000 17 clue sudokus, and I'm not going to look for it. It might start into another year-long obsession, and like I said: my friends already make fun of me for programming with sudoku too much!
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Re: Symmetric 18s

Postby ronk » Tue Oct 25, 2011 12:54 pm

eleven wrote:To give an impression of the variety of equivalent shapes i randomly selected 108 out of the 2244 (symmetry-keeping) equivalents of "#1 mauricio". So it would take some time, if you want to select a favorite shape for all representatives.
http://imageshack.us/photo/my-images/545/mauricio1.jpg/

I recommend picking the row-minlex version of each vertically symmetric 18-clue pattern, i.e., without the influence of clue values. This method removes arbitrariness of personal choices. Here are the sorted row-minlex patterns for the 106 known patterns. If anyone is interested, I can make another pass ... adding clues and preserving the origin.

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.............x......x...x.............xx.xx..xx.....xx...x.x....x.....x..x..x..x.
.............x......x.x.x............x.....x..x.x.x.x...x...x....x...x...x.x.x.x.
.............x......xx.xx.....x.x....x.....x..x.....x...x...x....x...x...x..x..x.
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............x.x.....x...x.............x...x...x.x.x.x.....x....x.......xxx..x..xx
............x.x.....x...x.............x...x...x.x.x.x.....x....x...x...xxx.....xx
............x.x.....x...x.............x.x.x...x.....x.....x.....x..x..x.xx..x..xx
............x.x.....x...x.............x.x.x...x.....x...x.x.x...x..x..x..x..x..x.
............x.x.....x...x.............x.x.x...x..x..x...........x.x.x.x.x.x...x.x
............x.x.....x...x.............x.x.x...x.x.x.x....xxx....x.....x..x.....x.
............x.x.....x...x............x.x.x.x.x.x...x.x....x......x.x.x...x.....x.
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............x.x.....x...x.....x.x.....x...x....x.x.x...x.....x..x..x..x.x.......x
............x.x.....x...x.....x.x....x.....x..x..x..x...x...x....x...x...x..x..x.
............x.x.....x.x.x..............x.x....xx...xx.....x....x.......xxx.....xx
............x.x.....x.x.x..............x.x....xx...xx...x.x.x..x.......xx.......x
............x.x.....x.x.x.............x...x...x.x.x.x............x.x.x..xx.....xx
............x.x.....x.x.x.............x...x...x.x.x.x.....x....x.......xxx.....xx
............x.x.....x.x.x.............x.x.x...x.....x...........x.x.x.x.x.x...x.x
............x.x.....x.x.x.............x.x.x...x.x.x.x.....x......x.x.x...x.....x.
............x.x.....x.x.x......x......x.x.x...x.....x.....x.....x.....x.xx.....xx
............x.x.....x.x.x......x.....x.....x..x.....x...x...x....x...x...x.x.x.x.
............x.x....xx...xx.............xxx...x.......x...........x.x.x..x..x.x..x
............x.x....xx...xx............x...x..x.......x....x......xxxxx..x.......x
............x.x....xx...xx............x...x..x.......x....x.....x.xxx.x.x.......x
............x.x....xx...xx............x...x..x..x.x..x...........x.x.x..x...x...x
............x.x....xx...xx............x.x.x..x.......x...........x.x.x..x..x.x..x
............x.x....xx...xx............x.x.x..x.......x....x......x...x..x..x.x..x
............x.x....xx...xx............x.x.x..x.......x....x....x.......xxx.....xx
............x.x....xx...xx.....x.......x.x.....x...x...x.....x.x.......xx...x...x
............x.x....xx...xx.....x....x.......xx.......x...x.x.....x...x....x.x.x..
............xxx.....x...x.............x...x...x.x.x.x.....x....x.......xxx.....xx
............xxx.....x...x.............x.x.x...x.....x.....x.....x..x..x.xx.....xx
............xxx.....x...x.............x.x.x...x.....x...x.x.x...x.....x..x..x..x.
............xxx.....x...x.............x.x.x...x.x.x.x.....x.....x.....x..x..x..x.
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............xxx.....x...x.....x.x....x.....x.x.......x..x...x....x...x...x..x..x.
...........x...x....x.x.x.....x.x......x.x.....x...x...x.....x..x..x..x.x.......x
...........x...x....xx.xx......x......x...x...x..x..x....x.x....x.....x..x.....x.
...........x...x....xx.xx......x......x...x...x..x..x....x.x....x.....x.x.......x
...........x...x...x..x..x............xx.xx...x.....x............xx.xx...x..x..x.
...........x...x...x..x..x............xx.xx...x.....x...........x..x..x.x.x...x.x
...........x...x...x..x..x............xx.xx...x.....x.....x....x.......xx.x...x.x
...........x...x...x..x..x............xx.xx...x.....x....x.x.....x...x...x..x..x.
...........x...x...x..x..x............xx.xx...x..x..x.....x.......xxx.....x...x..
...........x...x...x..x..x............xx.xx...x..x..x.....x......x...x...x..x..x.
...........x...x...x..x..x............xx.xx...x..x..x.....x.....x..x..x.x.......x
...........x...x...x..x..x............xxxxx...x.....x.....x.......xxx.....x...x..
...........x...x...x..x..x...........x.....x.x..x.x..x....x......x...x..x.x...x.x
...........x...x...x..x..x...........x.....x.x..x.x..x....x....x.......xx.x...x.x
...........x...x...x..x..x.....x.......x.x....x.....x.....x......x.x.x..x.x...x.x
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...........x...x...x..x..x.....x.......x.x....x.....x...xx.xx...x.....x..x.....x.
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...........x...x...x..x..x.....x......x...x...x..x..x.....x......xx.xx..x.......x
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...........x...x...x..x..x....x.x.....x...x....x.x.x.....x.x....x.....x.x.......x
...........x...x...x..x..x....x.x.....x...x..x...x...x...x.x....x.....x..x.....x.
...........x...x...x.x.x.x............x.x.x...x..x..x.....x......x.x.x...x.....x.
...........x...x...x.x.x.x............x.x.x..x..x.x..x....x......x...x...x.....x.
...........x...x...x.x.x.x............x.x.x..xx.....xx....x.......x.x.....x...x..
...........x...x...x.x.x.x.....x.......x.x.....x...x......x.....x.....x.xx.....xx
...........x...x...x.x.x.x.....x.......x.x.....x...x......x....x.......xxx.....xx
...........x...x...x.x.x.x.....x......x...x...x.....x.....x.....x.....x.xx.....xx
...........x...x...x.x.x.x.....x......x...x...x.....x.....x....x.......xxx.....xx
...........x...x...x.x.x.x.....x......x...x...x.....x....x.x.....x...x....x.x.x..
...........x...x...x.x.x.x.....x......x.x.x...x.....x....x.x.....x...x...x.....x.
...........x...x...x.x.x.x.....x.....x.....x..x..x..x....x.x.....x...x....x...x..
...........x...x...x.x.x.x.....x.....x.....x..x..x..x....x.x.....x...x..x.......x
...........x...x...x.x.x.x.....x.....x.....x.x.......x...x.x.....x...x....x.x.x..
...........x...x...xx...xx.....x......x...x..x...x...x...x.x...x.......xx.......x
...........x.x.x...x..x..x....x.x.....x...x....x...x.....x.x....x.....x..x.....x.
...........x.x.x...x.x.x.x.....x.......x.x.....x...x......x......x.x.x...x.....x.
...........x.x.x...x.x.x.x.....x.......x.x.....x...x......x.....x.....x..x..x..x.
...........x.x.x...x.x.x.x.....x.....x.....x..x.....x....x.x.....x...x....x...x..
...........x.x.x...x.x.x.x.....x.....x.....x..x.....x....x.x.....x...x..x.......x
...........xx.xx..xx.....xx....x........x......x...x.....x.x....x.....x..x.....x.
...........xx.xx..xx.....xx....x........x......x...x.....x.x....x.....x.x.......x
....x........x......x...x......x......xx.xx...x.....x....x.x....x.....x..x..x..x.
....x........x......x...x.....x.x....x.....x..x.....x...x...x....xx.xx...x.....x.
....x.......x.x.....x...x......x......x...x...xx...xx....x.x....x.....x.x.......x
....x.......x.x.....x...x.....x.x....x.....x..x.....x...x...x....x.x.x...x.....x.
....x.......xxx.....x...x......x......x...x....x.x.x.....x.x....x.....x..x.....x.
....x.......xxx.....x...x......x......x...x...x..x..x....x.x....x.....x..x.....x.
....x......x...x....x...x.....x.x......x.x....x.....x...x...x...x.....x..x..x..x.
....x......x...x....x...x.....x.x.....x...x...x..x..x....x.x....x.....x..x.....x.
....x......x...x...x.....x.....x......x...x..x.x...x.x...x.x....x.....x..x.....x.
....x......x...x...x.....x....x.x.....x...x....x...x.....x.x....x.....x..x..x..x.
....x......x...x...x.....x....x.x...x.......xx.......x...xxx.....x...x....x...x..
....x......x...x...x..x..x.....x......x...x...x..x..x....x.x.....x...x...x.....x.
ronk
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Re: Symmetric 18s

Postby eleven » Tue Jan 24, 2012 8:46 am

[Edit:] Picked the wrong puzzle :(
Last edited by eleven on Wed Jan 25, 2012 10:02 am, edited 1 time in total.
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Re: Symmetric 18s

Postby Serg » Tue Jan 24, 2012 9:22 pm

Withdrawn
Last edited by Serg on Mon Jan 30, 2012 1:07 pm, edited 1 time in total.
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Re: Symmetric 18s

Postby eleven » Tue Jan 24, 2012 9:32 pm

Oops thanks,
the pattern equivalence is easy to see with the naked eye.

Strange, my program told me that i would have 107 different patterns. Maybe a bug, maybe i copied the wrong puzzle.
I will see tomorrow.
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Re: Symmetric 18s

Postby eleven » Wed Jan 25, 2012 10:02 am

Here is the right one now, sorry.
Code: Select all
 +-------+-------+-------+
 | . . . | . . . | . . . |
 | . . 3 | . . . | 5 . . |
 | 6 . . | 7 . 9 | . . 1 |
 +-------+-------+-------+
 | . . . | . . . | . . . |
 | 9 . . | . 4 . | . . 7 |
 | 1 . . | . 5 . | . . 9 |
 +-------+-------+-------+
 | . . . | 6 . 1 | . . . |
 | . . 5 | . . . | 3 . . |
 | . 4 . | . . . | . 8 . |
 +-------+-------+-------+

...........3...5..6..7.9..1.........9...4...71...5...9...6.1.....5...3...4.....8. #107 eleven
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Re: Symmetric 18s

Postby Serg » Wed Jan 25, 2012 9:28 pm

Congratulations, eleven!
I confirm that this posted pattern 107 is new and unique. Well done!
eleven wrote:
Code: Select all
 +-------+-------+-------+
 | . . . | . . . | . . . |
 | . . 3 | . . . | 5 . . |
 | 6 . . | 7 . 9 | . . 1 |
 +-------+-------+-------+
 | . . . | . . . | . . . |
 | 9 . . | . 4 . | . . 7 |
 | 1 . . | . 5 . | . . 9 |
 +-------+-------+-------+
 | . . . | 6 . 1 | . . . |
 | . . 5 | . . . | 3 . . |
 | . 4 . | . . . | . 8 . |
 +-------+-------+-------+

...........3...5..6..7.9..1.........9...4...71...5...9...6.1.....5...3...4.....8. #107 eleven

Did you mean something or somebody when you selected representation of this pattern? As to me this pattern resembles Cheburashka - well-known in Russia the hero of several very popular cartoons.

Serg
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Re: Symmetric 18s

Postby eleven » Thu Jan 26, 2012 11:11 am

Serg wrote:Congratulations, eleven!
I confirm that this posted pattern 107 is new and unique. Well done!

Thanks again, Serg.

Did you mean something or somebody when you selected representation of this pattern? As to me this pattern resembles Cheburashka - well-known in Russia the hero of several very popular cartoons.

Nice, i did not know it. I just looked through some equivalent patterns and took this, which reminded me on a (headless) butterfly.

This is a picture with the patterns in minlex order, like Ron suggested (hopefully i calculated them correctly).
http://imageshack.us/photo/my-images/403/sym18.jpg

The new one is nr 6 in row 9, with a nice pattern too.

Here are all (385 - [edit:] 387) 18 clues with reflection symmetry i know, with the patterns in the same order. After the pattern you can find its number in olimpia's list. You can see, that for about the half of the patterns only one puzzle is known, on the other hand there is a pattern with 49 puzzles.

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...........xx.xx..xx.....xx....x........x......x...x.....x.x....x.....x.x.......x # 42
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Last edited by eleven on Sat Feb 04, 2012 4:52 pm, edited 1 time in total.
eleven
 
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Re: Symmetric 18s

Postby Serg » Thu Feb 02, 2012 9:11 am

Hi, eleven!
Thank you for posting interesting data.
Did you check all these 385 puzzles for minimality? It is (theoretically) possible to produce new 17-clue puzzles from non-minimal 18-clue puzzles.
What methods do you use to search for new symmetric 18-clue puzzles? Did you search for another valid puzzles in cases when other people posted new 18-clue sym. puzzles (patterns)?

Serg
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Re: Symmetric 18s

Postby eleven » Sat Feb 04, 2012 4:50 pm

Hi Serg,

Serg wrote:Did you check all these 385 puzzles for minimality? It is (theoretically) possible to produce new 17-clue puzzles from non-minimal 18-clue puzzles.

Only theoretically i guess. If i remember right (the posts were lost by the forum crash) ano1 has found out, that the big majority of the known 17's comes from a huge {-1+1} cluster of 800 mio 18's (i.e. from one puzzle in this cluster you can get to each other with a sequence of 18 clues, where the next puzzle - in appropriate transformation - only differs in single clue). Now these new 18's (70 of the patterns were in ano1's set) are very isolated puzzles, you only get a few other 18's with {-1+1}. But the smaller a cluster the worse are the chances to find a 17 in the near. Note, that when you add single clues to a 17, you already get a {-1+1} cluster with 64 (non minimal) 18's.
What methods do you use to search for new symmetric 18-clue puzzles? Did you search for another valid puzzles in cases when other people posted new 18-clue sym. puzzles (patterns)?

I extended the neighbourhood search a bit, which i already had used before. E.g. i made a {-5+5} for the known puzzles inside their pattern, and a {-4+4} to other symmetrical patterns in normal form (i.e. with c5 being the symmetry axis). For some puzzles i also made {-6+6} pattern searches and searches to other patterns replacing up to 6 cells. This took about 2 weeks, until i found the new pattern. Since then i only could find 2 more puzzles in pattern #8 (i add them to the list above) - and i will stop the search, when the current run will be finished tomorrow.
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Re: Symmetric 18s

Postby tarek » Sat Feb 04, 2012 7:48 pm

Is there a chance that the distance between patterns can help us find puzzles?

Choose 2 patterns: one with a known puzzle and one without. Shuffle one pattern so that it matches -as much as possible - the other pattern. Fill the matching positions with clues from a valid puzzle. Use gsf's program to search the remaining unfilled pattern clues (a feature in his program targeting the patterns game mentioned in the strategies section)

I know that gsf's program can measure the distance between puzzles, not sure if it can do it for patterns and show pattern transformations.

Tarek
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